Compute the following Riemann-Stieltjes integral: The integral from 0 to 3 of $dg(x)/(sqrt(1 + x^2))$ where $g(x) = |2x - 2| + |3x - 6| - |x - 3|$
What I have tried:
Using formula: sum of $f(c_i)*(g(x_(i+1)) - g(x-i))$. --> $1/\sqrt(2)*(1 - 5) + 1/\sqrt(5)*(1 - 1) = -2\sqrt(2)$ I'm not sure that this is the correct formula to use, but I saw that another person used it on this site solving a similar question. I'm not sure if my answer is correct or not.
Redefining the function g as follows
\begin{align} g\left( x \right) &= \left| {2x - 2} \right| + \left| {3x - 6} \right| - \left| {x - 3} \right| \\ &= \left\{ \begin{array}{l} 2x - 2,\,\,\,\,\,x \ge 1 \\ 2 - 2x,\,\,\,\,\,0 < x < 1 \\ \end{array} \right. + \left\{ \begin{array}{l} 3x - 6,\,\,\,\,\,x \ge 2 \\ 6 - 3x,\,\,\,\,\,0 < x < 2 \\ \end{array} \right. - \left\{ \begin{array}{l} x - 3,\,\,\,\,\,x \ge 3 \\ 3 - x,\,\,\,\,\,0 < x < 3 \\ \end{array} \right. \\ &= \left\{ \begin{array}{l} 2x - 2 + 3x - 6 - x + 3,\,\,\,\,\,\,\,x > 3 \\ 2x - 2 + 3x - 6 - 3 + x,\,\,\,\,\,\,\,2 < x \le 3 \\ 2x - 2 + 6 - 3x - 3 + x,\,\,\,\,\,\,\,1 < x \le 2 \\ 2 - 2x + 6 - 3x - 3 + x,\,\,\,\,\,\,\,0 \le x \le 1 \\ \end{array} \right. \\ &= \left\{ \begin{array}{l} 4x - 5,\,\,\,\,\,\,\,x > 3 \\ 6x - 11,\,\,\,\,\,\,2 < x \le 3 \\ 2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 < x \le 2 \\ 5 - 4x,\,\,\,\,\,\,\,0 \le x \le 1 \\ \end{array} \right. \end{align} then \begin{align}\int_0^3 {\frac{{dg\left( x \right)}}{{\sqrt {1 + {x^2}} }}} &= \int_0^1 {\frac{{dg\left( x \right)}}{{\sqrt {1 + {x^2}} }}} + \int_1^2 {\frac{{dg\left( x \right)}}{{\sqrt {1 + {x^2}} }}} + \int_2^3 {\frac{{dg\left( x \right)}}{{\sqrt {1 + {x^2}} }}} \\ &=\int_0^1 {\frac{{d\left( {5 - 4x} \right)}}{{\sqrt {1 + {x^2}} }}} + \int_1^2 {\frac{{d\left( 2 \right)}}{{\sqrt {1 + {x^2}} }}} + \int_2^3 {\frac{{d\left( {6x - 11} \right)}}{{\sqrt {1 + {x^2}} }}} \\ &= - \int_0^1 {\frac{{4dx}}{{\sqrt {1 + {x^2}} }}} + 0 + \int_2^3 {\frac{{6dx}}{{\sqrt {1 + {x^2}} }}} \end{align} Finally using the substitution $x = \tan u$, then we get the desired result.
